HAL Id: hal-00691603https://hal-enpc.archives-ouvertes.fr/hal-00691603

Submitted on 26 Apr 2012

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Unified semi-analytical wall boundary conditions forinviscid, laminar or turbulent flows in the meshless SPH

methodMartin Ferrand, Dominique Laurence, Benedict Rogers, Damien Violeau,

Christophe Kassiotis

To cite this version:Martin Ferrand, Dominique Laurence, Benedict Rogers, Damien Violeau, Christophe Kassiotis. Uni-fied semi-analytical wall boundary conditions for inviscid, laminar or turbulent flows in the meshlessSPH method. International Journal for Numerical Methods in Fluids, Wiley, 2013, 71 (476-472),pp.Online. 10.1002/fld.3666. hal-00691603

❯♥ s♠♥②t ♦♥r②

♦♥t♦♥s ♦r ♥s ♠♥r ♦r tr♥t

♦s ♥ t ♠sss P ♠t♦

rr♥ r♥ ♦rs ❱♦ ss♦ts

♠ ♠rt♥rr♥r

②

♥tr♦t♦♥

rt♥ ♦♥r② ♦♥t♦♥s s ♦♥ ♦ t ♠♦st ♥♥ ♣rts ♦ t♠♦♦t Prt ②r♦②♥♠s P ♠t♦ ♥ ♠♥② r♥t ♣♣r♦s ♥ r♥t② ♦♣ s ❬ ❪ rt ♦♥r② ♦♥t♦♥s r ss♥t s♥ ♥ ♠♥② ♣♣t♦♥s r ♣rs ♦♥ ♦♥ ss rqr s s ♦rs ♦♥ ♦t♥ ♦s ♦r s♦r♥ strtrs t♥ s♥ ①♥s strtr ♥trt♦♥s ♥ ♣♦r♣♥ts t t s ♥♦♦s ♣rrqst t♦ ♠♣r♦ tr♥ ♠♦♥ ♥r s

♥② ♠t♦s ♦r ♠♣♠♥t♥ s♦ s ♥ P ♥ ♦♣♦r t ♣st t♦ s ♠♦♥ t ♠♦st ♣♦♣r ♥ ♦♠♠♦♥② s ♥tr r♦ t♦rs

♣s ♦rs s s ♥♥r♦♥s ♣♦t♥t ❬❪

tt♦s ♣rts ♦r tr♥t② ♦st ♣rts t ♠♣t②r ♦ t r♥ s♣♣♦rt ♥ ♦♥r② t rt ♣rts t♣rsr ♣②s q♥tts s s ♣rssr ♥ ♦t② t♦ ♥♦r♥♦ s♣ ♦r r s♣ ♦♥t♦♥

♠♥②t ♦♥r② ♦♥t♦♥s s ♦♥ rt♦♥ ♦r♠t♦♥♥tr♦ ② sr♠ t ❬❪ r r♥♦r♠③t♦♥ ♦ tqt♦♥s s ♠ t rs♣t t♦ t ♠ss♥ r ♦ t r♥ s♣♣♦rt❲ sr r♥t ♦ ts ♠t♦ ♥ ➓ r ♥tr♥s r♥t

♥ r♥ ♦♣rt♦rs r ♠♣♦② tt ♥sr ♦♥srt♦♥ ♣r♦♣rts

♦ ts ♠t♦s ♥ts ♥ rs ♥♥r♦♥s♣♦t♥t ♦r ♦r♥② srs t ♥trt♦♥ t♥ ♣rs ♦ t♦♠s ♥s s t♦ ♠♦ t r♣s♦♥ t♥ ♣rt ♥ ♦♥r② ♣rts ♠t♦ s s② t♦ ♠♣♠♥t ♥ ♦r ♦♠♣① ♦♠trs ♥ ♦♠♣tt♦♥② ♣ ♦r t s t♦ s♣r♦s ♦r ♦r ♥st♥ t s

♠♣♦ss t♦ ♠♥t♥ ♣rts ① ♦♥ rt ♥ t ♣rs♥ ♦rt②

tt♦s ♣rts r♦♠♠♥ ♥ ❬❪ ♣r♥ts s ♥♦♥♣②s♦r ♦r t ♣♦st♦♥♥ ♦ ♦st ♣rts ♥ ♦♠♣① ♦♠trs♥ ♣rtr② ♥② ♣rtr② ♥ 3D ♦r♦r t ♦♠♣tt♦♥♦rt rqr s ♥♦t ♥ ♥ tt ♥rs t ♥♠r ♦ ♣rtst♦ t ♥t♦ ♦♥t ♥ t srt s♠♠t♦♥s ♥

♥② t s♠♥②t ♣♣r♦ s ttrt t♥s t♦ ts rt♦♥rt♦♥ ♠♥s tt s♦♠ ♣②s q♥tts s s ♠♦♠♥t♠ t♦♠t② ♦♥sr ❯♥♦rt♥t② t ♦r♥ tt♠♣t ♥♦t ♣rs♥t r ♥ s♠♣ ② t♦ ♦♠♣t r♥♦r♠③t♦♥ tr♠s ♥tr♦ rtr♠♦r t ♦r♠t♦♥ ♣r♦♣♦s s ♥♦t t♦ r♣r♦ ②r♦stt ♣rssrs ♦r t♦ t ♥t♦ ♦♥t t sr strss ♦♥

s ♣♣r tr♦r ts t s♠♥t② ♣♣r♦ ♦ ❬❪ ♥ ①t♥st s♦ tt t r② ♦ t ♣②s s s t ♣rssr ♥①t t♦ ss ♦♥sr② ♠♣r♦ ♥ t ♦♥sst♥t ♠♥♥r ♦♣ ♦r ♦rrt♦♣rt♦rs ♦s s t♦ ♣r♦r♠ s♠t♦♥s t tr♥ ♠♦s s ♦r ♣rs♥t tr ② ♥s

• t♠ ♥trt♦♥ s♠ s ♦r t ♦♥t♥t② qt♦♥ rqrs ♣rtr tt♥t♦♥ ♥ s r② ♠♥t♦♥ ② ❱ ❬❪ ♣r♦ tr s♥♦ ♣♦♥t ♥ s♥ ♣♥♥ ♥ t♠ ♦ t ♣rts ♥st② ♥♦ r♥r♥t ♦rrt♦♥s r s ② s♥ ♥r♦♥r② r♥♦rrt rs♦♥ ♦ t t♠ ♥trt♦♥ s♠ ♦ t ♦r♠ ♣r♦♣♦s ♥❬❪ ♦♥t♠ s♠t♦♥s ② st ♦r tr♥t ♦s ♥ t ♦♥t①t♦ rt ♦♥r② ♦♥t♦♥s r ♣♦ss

• ♦ ♦♠♣t t r♥ ♦rrt♦♥ ♠♥ ♥ ♦♥t ❬❪ s ♥ ♥②t s ♦♠♣tt♦♥② ①♣♥s rs sr♠ t

❬❪ ♥ t ❬❪ s ♣♦②♥♦♠ ♣♣r♦①♠t♦♥ ♥ t t♦ ♥ ♦r ♦♠♣① ♦♠trs ❲ ♣r♦♣♦s r t♦ ♦♠♣tt r♥♦r♠st♦♥ tr♠ ♦ t r♥ s♣♣♦rt ♥r s♦ t ♥♦t♠ ♥trt♦♥ s♠ ♦♥ s ♥② s♣ ♦r t ♦♥r②

• ♦♥r② tr♠s ss r♦♠ t ♦♥t♥♦s ♣♣r♦①♠t♦♥ r ♥② sr s♠♠t♦♥s ♦♥② rqr ♥♦r♠t♦♥ r♦♠ ♠s ♦ t ♦♥r② t♥q ♦♣ r ♦s s t♦ ♦rrt t♣rssr r♥t ♥ s♦s tr♠s ♥ ♥ ♣r♦ ♣②s② ♦rrtsr strss s♦ tt ♥ t s♦♥ qt♦♥ ♦ sr q♥tt②♥ s♦ rt② s♥ P s s t tr♥t ♥t ♥r②♦r ts ss♣t♦♥ ♥ k − ǫ ♠♦ ♦ tr♥

s ♣♣r s ♦r♥s s ♦♦s ♥ t ♥①t t♦♥ ♥tr♦ t ♦r♠t♦♥s ♦r ② ♦♠♣rss P ♥tr♦♥ t s srtst♦♥s ♦r♦♣rt♦rs s♦s ♦rs ♥ tr♥ ♠♦♥ ♥ t t♦♥ ♦♦♥ t♥ ♦♣ t ♦♥sst♥t ♦♥r② ♦♥t♦♥s t ♠♣r♦ t♠ ♥trt♦♥ ♥ ♣♦st♦♥♣♥♥t t♥q t♦ ♦♠♣t t ♥st② ♣♣rt♥ ♣rs♥ts t ♦♠♣tt♦♥ ♦ t r♥♦r♠③t♦♥ tr♠s s♥ ♥♦ t♠♥trt♦♥ s♠ ♦r ♣rs♥t♥ t ♥♥ ♦r ♥ t t♦♥ ♦♥♥♠r rsts

s P ♦r♠t♦♥s ♦r ② ♦♠♣rss

t♦♥♥

♦♥srt ♦r♥♥ qt♦♥s

st② ♦♠♣rss t♦♥♥ s ♠♦ ② st ♦ ♣rts ♥♦t ② t ssr♣ts (.)a ♥ (.)b ♥ ♦♠♥ Ω st ♦ t ♣rts s ♥♦t ② F r ♣rt a ∈ F ♣♦sssss ♥♦r♠t♦♥ ss ts ♠ss ma ss♠ ♦♥st♥t ts ♣♦st♦♥ ra ts ♦t② ua t r♥♥ rt ♦ t ♣♦st♦♥ ts ♥st② ρa ts ♦♠ Va = ma

ρa

♥ ts♣rssr pa s♣t srt③t♦♥ s s ♦♥ t♥ ♥tr♣♦t♦♥ ♦rr♥ ♥t♦♥ w, t ♦♠♣t s♣♣♦rt Ωa t♥ rrs t♦ t s♣♣♦rt ♦ tr♥ ♥t♦♥ ♥tr ♦r ra ♦ rs R ❲ ♥r② ♥♦t ② t ssr♣ts(.)ab t r♥ ♦ q♥tt② t♥ t ♣♦st♦♥s a ♥ b ♦r ♥st♥uab ≡ ua − ub ♥ rab ≡ ra − rb ♦ ①♣t♦♥s r ♠ t t ♦♦♥♥♦tt♦♥s wab ≡ w (rab) ♥ ∇wab ≡ ∇aw (rab) r t s②♠♦ ∇a ♥♦tst r♥t t t ♣♦♥t ra

❲t ts ♥♦tt♦♥s ♦♠♠♦♥② s ♦r♠ ♦t ♦♥t♥t② qt♦♥ s s ❬❪

dρa

dt=∑

b∈Fmb∇wab.uab

r ddt

♥♦ts t r♥♥ rt tt s t♦ s② t rt ♦♥t ♣rt ♣t t ♥ r r♦♠ t ♦♦♥ ♥t♦♥ ♦ t ♥st②

ρa =∑

b∈Fmbwab

♥s ♠♦♠♥t♠ qt♦♥ ♥ rtt♥ s ♦♦s

dua

dt= −

∑

b∈Fmb

(pa

ρ2a

+pb

ρ2b

)∇wab + g

r g s rt② qt♦♥ ♦ stt ♥s ♥st② ♥ ♣rssr

pa =ρ0c

20

γ

[(ρa

ρ0

)γ

− 1

]

r ρ0 s t rr♥ ♥st② ♦ t c0 s t s♣ ♦ s♦♥ ♥ γ = 7s ♥r② ♦s♥ ♦r tr

❱s♦s ♦rs

s♦s tr♠ s s ♥ ts ♦r ♥ P ♦r ♦♠♣t♥ t s♦s tr♠1

ρ∇. (µ∇u) s t♦ ♦rrs t s ❬❪

1

ρa

∇. (µ∇u)a =∑

b∈Fmb

µa + µb

ρaρb

uab

r2ab

rab.∇wab

t r♥t ♦♣rt♦r s ♥ ② ∇a ≡ ex∂

∂xa

+ ey∂

∂ya

+ ez∂

∂za

(ex, ey , ez) ♥

t ss t♦r tr ♦ t rts♥ ♦♦r♥t s②st♠ ♥ 3D

r t ②♥♠ s♦st② µ s ♥ ②

µ ≡ νρ

♥ ν s t ♥♠t ♠♦r s♦st②

♣rt♦r ♥t♦♥s ♦r tr ♥②ss ♥ srt ♦♣rt♦rs r♥t GradaAb r♥ DivaAb ♥ ♣♥ Lapa (Bb, Ab)♦ rtrr② srt sr Ab ♥ Bb ♦r t♦r s Ab s

GradaAb ≡ ρa

∑

b∈Fmb

(Aa

ρ2a

+Ab

ρ2b

)∇wab

DivaAb ≡ − 1

ρa

∑

b∈FmbAab.∇wab

Lapa (Bb, Ab) ≡ ρa

∑

b∈Fmb

Ba + Bb

ρaρb

Aab

r2ab

rab.∇wab

r Aab ≡ Aa − Ab ② r ♣♣r♦①♠t♦♥s ♦ t ♦♥t♥♦s r♥tr♥ ♥ ♣♥ ♦♣rt♦rs rs♣t② ♥♦t ② ∇a ∇. ♥ ∇.∇s t qt♦♥ ♦ ♦♥t♥t② ♥ t ♠♦♠♥t♠ qt♦♥ ♥ rrtt♥ s ♦♦s

dρa

dt= −ρaDivaub

dua

dt= − 1

ρa

Gradapb + g +1

ρa

Lapa (µb, ub)

♦♣rt♦rs Grada ♥ Diva r s t♦ s♦♥t t s ♣♦ss t♦♥ r♥ts ♦ ts tr ♦♣rt♦rs ♦♥sr♥ ts ♣r♦♣rt② ♦ ♥t♦♥s ❬❪

r♥ ♠♦♥ ♥ P

②♥♦s ♣♣r♦ ♦♥ssts ♥ ♦♥sr♥ ♦♥② t ♠♥ ♣rt ♥♦t ②u ♦ t ♦t② u ♥ t qt♦♥ ♦ r t♦s t♥ ♠♦♥ tts ♦ t tt♥ ♣rt ♦ t ♦t② ♦♥ t ♠♥ ♦t②

k− ǫ tr♥ ♠♦ ♦♣ ② ♥r t ❬❪ t♦ tr♥s♣♦rt qt♦♥s ♦ k t tr♥t ♥t ♥r② ♥ ǫ ts ss♣t♦♥ t♦ t♠♦♠♠t♠ qt♦♥ s ♠♦ s ♦♦s

Dρa

Dt= −ρaDivaub

Dua

Dt=

1

ρa

Gradapb + 23ρkb + Lapa (µb + µTb, ub) + g

Dka

Dt=

1

ρa

Lapa

(µb +

µTb

σk

, kb)

+ Pa − ǫa

Dǫa

Dt=

1

ρa

Lapa

(µb +

µTb

σǫ

, ǫb)

+ǫa

ka

(Cǫ1Pa − Cǫ2ǫa)

♥ t st♦st ♣♦♥t ♦

r t rtD

Dt≡ ∂

∂t+ u.∇ s t r♥♥ rt ♦♥ t

②♥♦s r ub k − ǫ ♠♦ ♥s t tr♥t ♥ts♦st② νT ≡ µT

ρt♦ t tr♥t ♥t ♥r② k ♥ ts ss♣t♦♥ ǫ ②

νTa = Cµ

k2a

ǫa

t ♦♥st♥ts σk σǫ Cǫ1 ♥ Cǫ2 ♥ ② ❬❪

♣r♦t♦♥ tr♠ ♦ k Pa s ♥ ②

Pa = νTaS2a

r S2a ≡ 2Sa : Sa s t sr ♠♥ rt♦str♥ t♥s♦r str♥ rt

♦ t ♠♥ ♦t② s ♥ ② Sa ≡ 12

(∇au + ∇au

T) ❱♦ ♥ ss

❬❪ srt③ ♥ t P ♦r♠ t ♦t② r♥t

Gradaub ≡ − 1

ρa

∑

b∈Fmbuab ⊗ ∇wab

♥ t ♦♦♥ ♦r s ♦ s♠♣t② r♦♣ ♦rrs t♦ ♥♦t

♦t② ♥ ♣rssr ♥ ts ♥♦t r♥♥ rtsd

dt rr

s♦ ♣ ♥ ♠♥ tt ♥r tr♥t ♦♥t♦♥s t ttr q♥tts r♦♥sr s ②♥♦sr ♥ t r♥♥ rt s ♦♥ t②♥♦s r ♦t②

♦♥r② ♦♥t♦♥s ♥ rt t♠

st♣♣♥

rt♦♥ ♦ ♦♥r② tr♠s s♥ ♦♥t♥♦s

♥tr♣♦t♦♥

sr♠ t s r♥♦r♠st♦♥

♥st ♦ ss♠♥ tt ρa ≃∑

b∈Fmbwab ♥rst♠ts ρa ♥ t

♣rt a s ♦s t♦ ♦♥r② s r sr♠ t ❬❪ r♥♦r♠s t st♠t♦♥ s♥ ♥t♦♥ γa

ρa ≃ 1

γa

∑

b∈Fmbwab

r γa s ♥ ②

γa ≡∫

Ω∩Ωa

w (r′ − ra) dV ′

♥ ♥r γa s ♥ r♥ ♣♥♥ ♦♥② ♦♥ t ♣♦st♦♥ ♦ t♣rt a t rs♣t t♦ ♦♥rs ♦ Ω r r♦♠ s♦ ♦♥r② γa = 1.

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b ab R

Ω

Ωa

γa

∂Ω

r r♥♦♥r② ♥trt♦♥

♥tr♦t♦♥ ♦ γa ♥t♦ t rt♦♥ ♦ t ♦r♥♥ qt♦♥s st♦ s♦♠ ② r♥s s s t ♥ ♦♥t♥t② qt♦♥ ❬❪ r strt♥r♦♠ s rrtt♥

dρa

dt=

1

γa

∑

b∈Fmb∇wab.uab −

ρa

γa

∇γa.ua

s ♥ ♦♠♣r t ♦r♥② t r♥t ∇γa s ♥ ②

∇γa ≡∫

Ω∩Ωa

∇aw (r′ − ra) dV ′ =

∫

∂Ω∩Ωa

w (r′ − ra)ndS′

r t s♦♥ ♥tr s ♦t♥ s♥ t ss t♦r♠ ♥ r n st ♥r ♦♥r② ♥♦r♠

♥ ♦rr t♦ t t ♥ ♥tr♥ ♦rs ♥ ♦♥tt ♦rs sr♠t ❬❪ r t ♥tr♥ ♥r② s♥ t qt♦♥ ♦ r♥ ② ♦t♥ ♥ ♥tr♥ ♦r t♦ t ♣rssr ♦rrt♦♥ ♦ sr♠ t

♥ ①♣rss ♥ tr♠s ♦ ♥ ♦♠♣t ♥ rt♦♥ s♥s ♦♣rt♦rsr♥t ♥ r♥ ♥ ♦♥ ♥ ♥

GradKa Ab ≡ ρa

∑

b∈Fmb

(Aa

γaρ2a

+Ab

γbρ2b

)∇wab −

Aa

γa

∇γa

DivKa Ab ≡ − 1

γaρa

∑

b∈FmbAab.∇wab +

1

γa

Aa.∇γa

s t ♦♣rt♦rs r② t s♦♥t♦♥ ♣r♦♣rt② ♥ rt♦♥s st ♦s ts ♣r♦♥ t ♦♥sst♥② t♥ t ♠♦♠♥t♠ qt♦♥ ♥ t ♦♥t♥t② qt♦♥ ♥ tr♠ ♦ GradK

a Ab s s② sr♠ t s ♦♥r② ♦r ♥ t ♦♦♥ ♣r♦♣♦s ♥♥♥ ♦r♠ ♦ t ttr ♠♦ ♥ ♠♦r rt r♣rs♥tt♦♥ ♦r♥ts ♦♥ t

♥r s♣ ♦ t ♦♥r②

♦r ♦♥rs ♦ rtrr② s♣ tr s♣ ∂Ω ♦ t ♦♠♥ Ω, s ♣♣r♦①♠t t strt s♠♥ts ♥ 2D ♥♦t ② t ssr♣t (.)s

b aR

e1

e2Ss

ns

♦♥r② t s♠♣rts e ♥ r♥ ♥ ts♠♥ts s sr Ss ♥ ♥ ♥r ♥♦r♠ns

e

θ

e e

t ♦ t ♦♠ ♦ ♥ ♣rt

r ♦♥r② ♣r♦♣rt② ♥t♦♥s

♥♦r♠ ns ♥ sr r Ss s r st ♦♥t♥♥ ts♠♥ts s ♥♦t ② S s♠♥t s ♥ ② t♦ ♣♦♥ts ♥♦t② t ssr♣t (.)e1 ♥ (.)e2 ♥ ♥t ♦♠ Ve ♥ ② Ve = me

ρ0

r ρ0 s rr♥ ♥st② ♥t ♦♠ ♦ ♣rts s rt♦♥♦ t ♥t ♦♠ ♦ ♣rts Vf ♦r ♥st♥ ♦r ♣♥ Ve = 1

2Vf

♦r ♥r② ♦r ♥ ♣rt ♦♥ t ♥ ♥ θ s♣② ♦♥t r Ve = θ

2πVf

st ♦♥t♥♥ t ♣rts s ♥♦t ② E s ♣rtss♦ s♠ ♣rts ♥ ts rt r ♦ ♣rtr ♥trst ♦r r♦r♥t ♣rssr t t s♦ ♦♥r② ♥ ♥ ♦r ♥ strtr ♦♣♥ ♦r ①♠♣ ② r s♦ s t♦ ♠♣r♦ r② ♦ t ♦♥t♥t②qt♦♥ s t② ♠♠ t t s ♠♣♦rt♥t t♦ ♥♦t tt t② rt♥ ♥t♦ ♦♥t ♥ t ♦♥t♥t② qt♦♥ ♥ ♥ t ♠♦♠♥t♠ qt♦♥ E ⊂ F ♥ t② r r♥ ♣rts tt s t♦ s② t② r ① t s ♠♦t♦♥ss ♥ ♦s ♥♦t ♣♥ ♦♥ t ♠♦♠♥t♠ qt♦♥

①t ♥ t ♦♥trt♦♥ ♦ t s♠♥t s ♥ t ♦ ∇γa t♦

∇γas ≡(∫

re2

re1

w (r) dl

)ns

♥ t♥ ∇γa ♥ ♦♠♣♦s ♥

∇γa =∑

s∈S∇γas

sr♣t♦♥ ♦ t ♦♥r② ♦♠tr② ♥ ①t♥ t♦ 3D ② ssttt♥ t s♠♥ts ② tr♥s ♥ ts s ts ♦♠ Ve ♦ ♣rts♦ s t s♦ ♥ ♦ t

❲ ♦rrt r♥ts

♠♥ s♥t ♦ ♣r♦s t♥qs t♦ t ∇γa s tt t r

♥t ♦♣rt♦r ♥ ② s ♥♦t rt ♥r ♦♥r② ♦r t rsr♥ ♥ s tt r♥ts ♦ ♦♥st♥ts r ♥♦♥ ③r♦ s♠t ♥♦r♠ ♦r♣rssr ♥ ♣r♦ ♣♣ t♦t ♥② ♦② ♦r t r♥t

♦ t ♣rssr s ♥♦t ③r♦ r②r ♥ ♣rts rrr♥ t♠ss sr ♦ ♦rrt tt ♥ ♦ t♦ t ♦♥t♥♦s ♥tr♣♦t♦♥♦ ♥ rtrr② ♥tr ♥t♦♥ f t ♣♦♥t r

〈f〉 (r) =1

γ (r)

∫

Ω∩Ωr

f (r′) w (r) dV ′

r r ≡ |r − r′| ♥ Ωr s t r♥ s♣♣♦rt ♥tr ♥ r ② ♥tr♣♦t♥t r♥t ♦ t ♥t♦♥ f ♥ t s♠ ② t ♦♠s

〈∇f〉 (r) = − 1

γ (r)

∫

Ω∩Ωr

f (r′) ∇w (r) dV ′

− 1

γ (r)

∫

∂Ω∩Ωr

f (r′) w (r)ndS′

r t rt♥s s ♦t♥ ② ♥ ♥trt♦♥ ② ♣rts ♥ n s t♥r ♥♦r♠ ♦ t ♦♠♥ t t ♣♦st♦♥ r′ ♥ ♥ s tt t♦♥r② ♦♥t♦♥s ♣♣r ♥tr② tr♦ t s♦♥ ♥tr ♦

♦r♦r ♦♥sr tt t r♥t ♥ ② s srt ♣♣r♦①♠t♦♥ ♦ t ♦♥t♥♦s r♥t ∇f ≡ ρ∇

fρ

+ fρ∇ρ t♦ ♦t♥ s②♠♠tr

♦r♠t♦♥ ♦t♥ s s♦ ❬❪

〈∇f〉 (r) ≃⟨

ρ(r)∇f

ρ+

f

ρ(r)∇ρ

⟩(r)

= − 1

γ (r)

∫

Ω∩Ωr

[f

ρ(r′) ρ (r) +

f

ρ(r) ρ (r′)

]∇w (r) dV ′

− 1

γ (r)

∫

∂Ω∩Ωr

[f

ρ(r′) ρ (r) +

f

ρ(r) ρ (r′)

]w (r)ndS′

♥ ♥ t ♦♥r② ♦♥t♦♥s ♣♣r ♥tr② ♥ ♥ ♥♦ stt ♦♦♥ srt ♦♣rt♦r r♥t ♦r ♥ rtrr② Ab s

GradaAb ≡ ρa

γa

∑

b∈Fmb

(Aa

ρ2a

+Ab

ρ2b

)∇wab −

ρa

γa

∑

s∈S

(Aa

ρ2a

+As

ρ2s

)ρs∇γas

♠t♦ t♦ ♦♠♣t ∇γas sss ♥ ➓ rs t ♦♠♣tt♦♥ ♦ ρs ♥ As ♥stt ♥ t ♣rr♣ ♦♥ ②♥♠ ♦♥r②♦♥t♦♥s ♦t tt t srt r♥t rs r♦♠ sr♠t s ♥ ② ♦♥② ♥①t t♦ ♦♥r② ♠♥s tt ♦♥srt♦♥♣r♦♣rts r st r r♦♠ t s

t s ♥♦ ♦rrt s♥ t s♠ t P r♥t ♦ t♦r ♣♣rs ♥ t ♦t② r♥t ♦r ♥st♥ ttr q♥tt②♣②s ② r♦ ♥ t k − ǫ tr♥ ♠♦ s t s rs♣♦♥s ♦r t ♣r♦t♦♥ ♦ ♥t ♥r② ♥ t t tt ♥ ♥♥ ♦ t str♥ rts t rst ♥ t ♥t② ♦ t ♦♥r② t s ♠♣♦rt♥t t♦ rt ♥ts r s ♦r♠ ♥s t♦ ♦rrt t rs♣t t♦ t ♦♥rs♥ tt t t♥s t♦ ♥rst♠t t str♥ rt ♥①t t♦ ♦ ♦rrttt ♣r♦♣♦s ♥ s♠r ② s

Gradaub = − 1

γaρa

∑

b∈Fmbuab ⊗ ∇wab +

1

γaρa

∑

s∈Sρsuas ⊗ ∇γas

X(m)

Z(m

)

0 0.05 0.1 0.15

0

0.05

0.1

0.15

0.2

P(Pa)

5500050000450004000035000300002500020000150001000050000

sr♠ t s r♥t ♦♣rt♦r s ♥ t♦ ♠♥t♥ ♥♦r♠ ♣rssr Prts ♠♦ t♦♥ ♥♦♥♣②s qr♠

X(m)

Z(m

)

0 0.05 0.1 0.15

0

0.05

0.1

0.15

0.2

P(Pa)

5500050000450004000035000300002500020000150001000050000

♣rs♥t r♥t ♦♣rt♦r s t♦ ♠♦st ♠♥t♥ ♥♦r♠ ♣rssr s ♥ qr♠

r ♦♠♣rs♦♥ ♦ r♥t ♦♣rt♦rs ♥ ♥ ♥ ♦r♣rssr③♣r♦ ♣♣

♥ t t ♦rrt ♦♠♣♦♥♥ts t str♥ rt S s ♦♠♣t ♦r♥t♦ t ♥t♦♥ ♥ ②

♣♥

♣♥ rst♥ r♦♠ t ♦rrs ♠♦ ♥ s♥ s srtst♦♥ ♦ t ♦♥t♥♦s ♥tr♣♦t♦♥ 〈∇.µ∇f〉 (r) r f s t ♦♥t♥♦ssr ♥ µ t ♦♥t♥♦s s♦st②

〈∆ (µ, f)〉 (r) ≡ 〈∇.µ∇f〉 (r) = 〈∇r′ . [(µ∇f) (r) + (µ∇f) (r′)]〉 (r)

= − 1

γ (r)

∫

Ω∩Ωr

[(µ∇f) (r) + (µ∇f) (r′)]∇w (r − r′) dV ′

− 1

γ (r)

∫

∂Ω∩Ωr

[(µ∇f) (r) + (µ∇f) (r′)] .nw (r − r′) dS′

♦♠♥ t t ♥t r♥ ♣♣r♦①♠t♦♥

∇f(r). (r − r′) ≃ (f(r) − f(r′)) ≃ −∇f(r′). (r′ − r)

s ♥ ➓ t ♦♥r② tr♠s ♣♣r ♥tr② ♥ t s♦♥ ♥ ♦ r♦♠ ♥ ♥trt♦♥ ② ♣rts ♥ n s t ♥r ♥♦r♠ ♦ t ♦♠♥t t ♣♦st♦♥ r′ ❲ ♦♥trt♦♥s ♥s ♥ t ♣♦st♦♥ r s r r♦♠t ♦♥r② s♥ t r♥ w s ♦♠♣t s♣♣♦rt s t ♣r♦♣♦s♦rrt ♣♥ ♦♣rt♦r s

1

ρa

Lapa (Bb, Ab) =1

γa

∑

b∈Fmb

Ba + Bb

ρaρb

Aab

r2ab

rab.∇wab−1

γaρa

∑

s∈S(Bs∇As + Ba∇Aa) .∇γas

s♠ s st ♦s t ♥ rtrr② t♦r Ab ♥ s s

t♦ ♦rrt t s♦♥ tr♠ ♥ t rt♦s qt♦♥

❲ ♦rrt ♦♣rt♦rs ♥ t srt rt♦s

qt♦♥s

②♥♠ ♦♥r② ♦♥t♦♥s ♦♥ t ♣rssr

r♥t ♦♣rt♦r ♣♣ t♦ t ♣rssr s

Gradapb ≡ ρa

γa

∑

b∈Fmb

(pa

ρ2a

+pb

ρ2b

)∇wab −

ρa

γa

∑

s∈S

(pa

ρ2a

+ps

ρ2s

)ρs∇γas

♥ r t t♦ ♦♠♣t t ♣rssr ps ♥ t ♥st② ρs ♦♥ t ♥♦s ♥ ♥ r r♦st ② t rst ♦rr ②♥♠ ♦♥t♦♥ s∂ρ∂n

= 0 s ♦♥t♦♥ s ♦♥sst♥t t t r♥♦r♠③t♦♥ ♦♥ ♥ qt♦♥ ♥ t ♣rs♥ ♦ rt② ♥ ♠♦t♦♥ t ♥♦♦s ♦♥t♦♥ ♦♥ t♣rssr s rtt♥ s

∂

∂n

(p⋆

ρ+

u2

2

)= 0

r p⋆ ≡ p − ρg.r ♥ u s t ♠♥t ♦ t ♦t②

♦ ♦♠♣t t ♣rssr ♥ t ♥st② t t ♥ P ♥tr♣♦t♦♥♥ s ♦r t ♣rts ♥ E t ♦ ♠ s t♦ r ♥ s♣t♦

ρe =1

αe

∑

b∈F\EVbρbwbe

pe

ρe

=1

αe

∑

b∈F\EVb

(pb

ρb

− g.rbe +u2

b − u2e

2

)wbe

r t st F \ E ♥♦ts ♣rts F ①♥ ♣rts ♥ E ♥r αe s ♥ ②

αe ≡∑

b∈F\EVbwbe

sr♣t♦♥ ♦ t ♣r tr αe s ♥ ♥ ➓ s qt♦♥

r t s ♠♣♦rt♥t t♦ ♥♦t tt t ♥tr♣♦t♦♥ ♦♥ s s ♦♥ ♣rts ♥ F ♦ ♥♦t ♦♥ t♦ t st ♦ ♣rts E ♦r s♠♣t② r♠♥ ♥ 2D ♥ ♥ t ♥st② ♥ t ♣rssr t t ♠♥ts st♦

ρs =ρe1 + ρe2

2ps

ρs

=pe1/ρe1 + pe2/ρe2

2

r t ♥♦s ♦r ♣rts e1 ♥ e2 r ♥ ♥ ➓ sstrt② t♦ t q♥tts t t rs t♦ ♦♠♣t t sr strss ♥ ➓ ♥ s♦ t ♦ sr tr♥s♣♦rt ② t ♦♥ ♣♣♥① r s ♥♦ t♦rt rstrt♦♥ t♦ t s♣♥ ♦ t ♥♦s

❲ sr strss

♦rrt ♦r♠ ♣♣ t♦ t u s

1

ρa

Lapa (µb, ub) =1

γa

∑

b∈Fmb

µa + µb

ρaρb

uab

r2ab

rab.∇wab−1

γaρa

∑

s∈S|∇γas| (µa∇ua + µs∇us) .ns

♥ s♦ s ♥ ♣ ♦ Lapa ♥

♦♥r② tr♠s r t♥ trt s♥ t rt♦♥ ♦t② uτ ♥②

µ∂u

∂n

∣∣∣∣wall

≡ ρuτuτ

r♣rs♥ts t sr strss t t ♥ t ♣rs♥t rt ② ♦♥♥t♦♥ uτ s ♦s♥ t♦ t s♠ rt♦♥ s t ♦ ♦t② s t♥r♣s (µ∇u)s .ns ♥ t ♦♥r② tr♠ ♦ s ♥ ss ♥t ♠♥t♦r ♥t ♦♠ rt♦s ♦s

(µ∇u)a .ns ≃ ρuτsuτs

♦♠♣tt♦♥ ♦ t rt♦♥ ♦t② ♥ ♠♥r s rt♦♥♦t② uτ s q♥tt② ♥ t t ♦♥r② ♦ ♦♠♣t t ♥ ♦♠♣tt♦♥ ②♥♠ ♦ ♥st ♦ s♥ ts ♥t♦♥ s②t t ♥t ♦ ♥♦♥ t ♣②s ♦r ♦ t ♦t② ♥ t♥t② ♦ t ♦♥r② ♦r ①♠♣ ♥ ♠♥r tst s t ♦t② ♣r♦ s ①♣t t♦ ♥r ♦s t♦ t ♥ t♥ t ♦♦♥ rt♦♥s♣t♥ st♥ t♦ t z ♥ ♦t② ♦♥ t u ♦s

uτuτ = limz→0

νu

z

♠♥ ♥t s tt ♦ ♥♦t ♥ t♦ st♠t t rt ♦t ♦t② ♥①t t♦ t r t s t t♦ ♦♠♣t ♥♦tr♥t s tt ♥ ①t♥ t ♥t♦♥ ♦ t rt♦♥ ♦t② ♥ t rr ♣rts ♥trr t t ♦♥r② tt s ♥ t r♥ s♣♣♦rt♥trsts t s ♥ ♥

uτauτa =νua

za

r za s t st♥ t♦ t ♦r ♣rt a

♥t② t♦ t ρsuτsuτs s ♥ t ♦♥t♥t② ♦ strsss t♦ sts

ρeuτeuτe =1

αe

∑

b∈F\EVbρbuτbuτbwbe

♥ rhosuτsuτs s t r t♥ t ♣rts e1 ♥ e2 ♥♥ ➓ s ♦r♠ r s♠r t♦

s♠ trt♠♥t s ①t♥ t♦ tr♥s♣♦rt qt♦♥ ♦ sr ss k ♦r ǫ ♥ t k − ǫ tr♥ ♠♦ ♥ t ♣♣♥①

♦♥srt♦♥ sss t♠ ♥trt♦♥ ♦r t ♦♥t♥

t② qt♦♥

♦r♥ t♠ ♥trt♦♥ s♠ s ♥ ♣r♦s ♦r ❬❪ ❬❪ s s♠♣rst♦rr s②♠♣t s♠ r ♥ ♠♣t ♦t② s s ♥ t♣t s ♦ t ♣♦st♦♥ ♥ ♥st②

un+1a = un

a − δt

ρna

Gradnapn

b + g

rn+1a = rn

a + δtun+1a

ρn+1a = ρn

a + δt∑

b∈Fmb∇

nwab.un+1ab

r t s♣rsr♣t (.)n rrs t♦ t t♠ st♣ n ♥ t♦ t t♠ t =

n∑

i=1

δt

♥ ts s♠♠♣t s♠ t ♦ts r ①♣t rs t ♣♦st♦♥sr ♠♣t ♥ t ♦♥t♥t② qt♦♥ ♣♦st♦♥s r ①♣t rs t ♦ts r ♠♣t ♦r ts rs♦♥ ♦ ♥♦t rt t rs ♦ qt♦♥ s ρaDiv ub

♠♣r♦♥ t t♠ ♥trt♦♥ ♦ t ♦♥t♥t② q

t♦♥

♦ ♣t t ♣r♦s t♠ ♥trt♦♥ s♠ t♦ t ♠t♦ ♦ sr♠t ♥ t ♣rs♥t ♠♦ ♦♥ t ♦♦♥ s♠ s ♣♦ss

un+1a = un

a − δt

ρna

Gradn

apnb + g

rn+1a = rn

a + δtun+1a

ρn+1a = ρn

a +δt

γna

[∑

b∈Fmb∇

nwab.un+1ab − ρn

a∇nγa.un+1

a

]

r t ♦♣rt♦r Grada s tr ♦r ①♣r♥ ♦ t t♦rss s♦♥ ts ♣♣r♦ s♠s t♦ stst♦r② rsts ♦r ♠ r s♥ s♥ r t s♦ s r ♣t ♠♣r♠ t rt② s♠t♠ st♣ ♦r ♥ r♥♥♥ ♦♥t♠ s♠t♦♥s ♥ ♥♥ t rt② r t♠ st♣ ♣rts ♥r t ♠♦ rt② ♦♥rss♦② ♥ ♥t② ♣ss tr♦ t ♦♥r② s s ♥ r rt② t♠ st♣ s st ② rs♥ t ♥♠r s♣ ♦ s♦♥ c0

♣r♦♠ s s ② t ♦♥t♥t② qt♦♥ ♥ ♣rts ♥r t♦♥r② r ♦st♥ ♠♦♥ ♥ ♦rt tr ♥sts rs♥ t♥ t ♣rssr rt t♦ t ♥st② ② t qt♦♥ ♦ stt ♦♠s♥s♥t t♦ rt r♣s ♦r t♦ ♥ t ♦tr ♦rs

♦r♥ ♦ ts ♣♥♦♠♥♦♥ s t tr♠ δtγn

a

ρna∇

nγa.un+1a ♥ t t♠

srt③ ♦♥t♥t② qt♦♥ ♥ ♦♥sr s♥ ♣rt♠♦♥ t♦rs t t♥ t t♠s tn ♥ tn+1 r♦♠ t st♥ zn

t♦ zn+1 t ①t rt♦♥ ♦ t ♥st② s ♥♦t r♣r♦ ② t srt

X(m)

Z(m

)

0 0.10

0.2

0.4

0.6

0.8

1

P(Pa)

100009000800070006000500040003000200010000

Pr♦s t♠s♠ tc0 = 20m.s−1

X(m)

Z(m

)

0 0.10

0.2

0.4

0.6

0.8

1

P(Pa)

100009000800070006000500040003000200010000

Pr♦s t♠s♠ tc0 = 100m.s−1

X(m)

Z(m

)

0 0.10

0.2

0.4

0.6

0.8

1

P(Pa)

100009000800070006000500040003000200010000

t♠s♠ tc0 = 20m.s−1

r ♦♠♣rs♦♥ ♦ t ♣rssr ♥ t tr ♣t ♥ ♣r♦♦♣♥ ♥♥ ♦♥ r♦♠ t t♦ rt ♦r t♦ r♥t t♠ s♠s tr ts♠ ♣②s t♠

♦r♠ ❬❪ s s②st♠t rr♦r ♦ t t♠ srt③t♦♥ ♦ t ♦♥t♥t②qt♦♥ s ♥sr

♥② ♦tr ♥trt♦♥ t♠ s♠s ♥ ♦♥sr s s ♣r♦t♠ s♠ ♦ r t rr♦rs ♥ t ♥trt♦♥ ♦ t ♦♥t♥t②qt♦♥ ♦ ♠ ♥ t ♣rs♥t ♦r s t♦ ♦♥sr ♥ ♣♣r♦ tts t ♥st② ①♣t② s ♥t♦♥ ♦ t ♣rts ♣♦st♦♥s s s ♠♦r r♦st ♣♣r♦ s♣② tr s r♣s ♦r ♥t♦♥ ♦ t♣rssr ♥ ♥ ♦ t ♥st② ♥ ♦♠♣rss ♦s

♦♠♣t② ♣♦st♦♥♣♥♥t ② t♦ ♦♠♣t t ♥st②

rtr♥ t♦ t ♠♥ ♦ ♦rrt♥ t ♥♦♠♣t r♥ s♣♣♦rt ♥ s tt t ♦rrt ♦♥t♥t② qt♦♥ ♦♠s r♦♠

d (γaρa)

dt=

d

dt

(∑

b∈Fmbwab

)

γa

dρa

dt+ ρa

dγa

dt=

∑

b∈Fmb∇wab.uab

♥ s♠r ② ❱ ❬❪ stt tt t ♦♥t♥t② qt♦♥ s strt②

q♥t t♦dρa

dt=

d

dt

(∑

b∈Fmbwab

) t♠ s ♦♥sr ♦♥t♥♦s r

qt♦♥ s s ② t♦ ♥trt ①t② ♥ t♠ t q♥tt② γaρa

t ♣rts ♣♦st♦♥s ♠♦ r♦♠ rnb t♦

rn+1

b

s ♣r♥ts s②st♠t

t♠ ♥trt♦♥ rr♦rs ♥ ♠s ρna ♣♥ ♦♥② ♦♥ t ♣♦st♦♥s ♦ ♣rts

t t s♠ t♠ ts ♣r♦♣rt② t♦tr t s♠♣t t♠st♣♣♥ ♥srst ♦♥srt♦♥ ♦ ♥ ♥r② s ❬❪ ♦r ts

s s t♦ t ♦♦♥ t♠ ♥trt♦♥ s♠

un+1a =un

a − δt

ρna

Gradn

apnb + g

rn+1a = rn

a + δtun+1a

(γaρa)n+1

=(γaρa)n

+∑

b∈Fmb

(wn+1

ab − wnab

)

r s♦s t rst ♦t♥ ② r♥ t t♠st♣ ♥ ♥t s②st♠t rr♦r ♥ t ♥st② qt♦♥ ♥ t t♠ s♠ t trst ♦t♥ t t t♠s♠ t rr t♠ st♣ ② ♦♥s♦ s♠ ♦ s ♦r t♦ st ♥ t♦ ♦♥r t t s r tt ♦st ♥st② r♥ t stst♦♥ t♠ t tr ♣t s rss rs t s ♥♦t t s ♦♥ t r t t ♥ s♠

♥t③t♦♥ ♦ t ♥st② t♠s♠ rqrs ♥ts ♦r t ♥st② ♥② ♦s r ♣♦ss rst ♦ ♣rt a♥ t rr♥ ♥st② ρ0 s ♥t

ρ0a = ρ0

s s ♦♥ ♣r♦s② t t ♦♥t♥t② qt♦♥ ♦ ♥ts tt t ♦♥t♥t② qt♦♥ ♦♥② ♠srs t rt♦♥ ♦ ♥st② ♥ ♥♦tt ♥t s♦rr ♦ t ♣rts ♦r t ♠♥ r s ♥ ♥♦♠♦♥t② t♥ ♣rts ♦ t♦ ♥♦♥♣②s ♦r s s ♣rts♦r♥② t t rsr rt♥ r♦♥ ♦ r♣s♦♥ ♥ srr♦♥ ②♦trs tr ♥ t s♠t♦♥ ♥ t ♥t ♥st②

ρ0

a

s ♥t③

♦r♥ t♦

ρ0a =

1

γ0a

∑

b∈F0

mbw0ab

s ♥t③t♦♥ s t ♥t t♦ ♠♥t♥ ♦♠♦♥t② t♥ ♣rtst t♥ rqrs rsr ♦rrt♦♥

rsr ♦rrt♦♥ γ ♦rrt♦♥ ♣rs♥t s♦ r ♦s ♥♦t t

♥t♦ ♦♥t ♥② rsr ♦rrt♦♥ r ρa ≡∑

b∈Fmbwab ♣♥s

♦♥② ♦♥ t ♣rts ♣♦st♦♥s s s ♦r ♦♥srt♦♥ ♣r♦♣rts ♣r♦♠ s tt ρa ♠srs t♦ r♥t q♥tts

t r♥ ♦ t ♣rts s q♥tt② ♦ ♥trst ♥

t ♣rs♥ ♦ ♦s t♥ t r♥ s♣♣♦rt ♦ ♣rt

❲ r t t♠ st♣ ② stt♥ t s♣ ♦ s♦♥ t 100m.s−1 ♥st ♦20m.s−1

t s rqr tt t ♦ s ♦rrt t γa ♥①t t♦ t ♥♦t ♥①tt♦ t rsr ♦ ts s t ♦♦♥ ♣r tr ♥②

α (r) =∑

b∈F

mb

ρb

w (r − rb)

s♦ tt ♦r ♣rt a ∈ F \ E

αa ≡∑

b∈F

mb

ρb

wab

♦r ♥ ♣rt e ∈ E ♥ ♦r t ♠ ♦ s♠♥t s ∈ S t♦ r♥t♥t♦♥s ♦ α r s

αe ≡∑

b∈F\E

mb

ρb

web

s♥ αe s s t♦ t q♥tts s s t ♥st② ♦r t ♣rssr tt ♥ ♦s♥ ♥♦t t♦ t ♥t♦ ♦♥t ♣rts ♥ ♥tr♣♦t♦♥ s♥ ♦♥② t ♣rts ♦ t ♣②s q♥tts s st ♥st② ρ ♦r t ♣rssr p t t s s ➓

♠ s t♦ ♣♣② t ♣r tr ♦♥ t ♥st② t t♠st♣ t ♦♥② ♥①t t♦ t rsr s tt t ♥st② s ♥♦t ♦rrtr②r t αa s ♥ t ♥t② ♦ rsr tr s ♦♥t♥♦s♠① t♦ ♦rrt t ♦♥t♥t② qt♦♥

ρa [βγa + (1 − β) αa] = ρa =∑

b∈Fmbwab

r

β = exp

[−K

(min

αa

γa

; 1

− 1

)2]

♥ K s t♥ t♦ ♥ rtrr② ♦ − ln(0.05)

0.012≃ 30000 s♦ tt

β ≤ 0.05 ♥ αa

γa

≤ 0.99 ♦t β s ♥ t sr♠rr ♥s t ts s ♠♦st ♦♥ rs t t♥s t♦ ③r♦ s ♣♣r♦ t rsr

♦♠♣tt♦♥ ♦ t r♥♦r♠③t♦♥ tr♠s

♦r♠ ♥t♦♥s ♦ t ♦♠tr q♥tts γa ♥ ∇γa ♦r ♣rta r

γa ≡∫

Ω∩Ωa

w (r − ra) dr

∇γa ≡∫

Ω∩Ωa

∇aw (r − ra) dr =

∫

∂Ω∩Ωa

w (r − ra)ndS

❲ r tt ∇γa r♣rs♥ts ♥ ♣♣r♦①♠t♦♥ ♦ t ♥♦r♠ t♦ t ♦r ♣rt ♦t t t ♣♦st♦♥ ra t ♣r♦s ♣♣r♦s s♥ ♣♦②♥♦♠♣♣r♦①♠t♦♥ ❬ ❪ ♥ ♥②t s♦t♦♥ ❬❪ ♥ srt s♠♠t♦♥ ♦r♦♥r② ♣♦♥ts ❬❪

s ♣♣r♦s ♥ts ♥ s♥ts sss rrr♥ ♦♠♣tt♦♥ ♦ t r♥♦r♠③t♦♥ tr♠ ♦ t r♥ s♣♣♦rt ♥r s♦ s ♦t♥ t t♠ ♥trt♦♥ s♠ tr② ♠♦r s②♦♥t♥ ♦r ♥② s♣ ♦ ♦♥rs ♣rs♥t ♥ ➓

♥②t ♦ ∇γa

❲t t ♦♥r② ♦ t ♦♠♥ ♦♠♣♦s ♦ s♠♥ts ♥♦t t t ssr♣t (.)s s♠♥t s ♥ ♥r ♥♦r♠ ns ♥♥♥ ♣♦♥t re1 ♥ ♥♥♥ ♣♦♥t re2 s r ♥ ♥ ♦♠♣t t ♥②t ♦ t ♦♥trt♦♥ ∇γas ♥ ②

∇γas ≡(∫

re2

re1

w (r) dl

)ns

r ♠ s♦♥ t ♥t♦♥s ♦ t ♦♠tr ♣r♠trs s t♦♦♠♣t t ♥②t ♦ ∇γas t

s ♦r t q♥t ❲♥♥ r♥ s ♥ ts ♦r

h

∫re2

re1

w (r) dl =(q2 cos α2)

πPq0

(q2) −(q1 cos α1)

πPq0

(q1)

+q40

π

(105

64+

35

512q20

)

sign (q2 cos α2) ln

(q2 + |q2 cos α2|

|q0|

)

−sign (q1 cos α1) ln

(q1 + |q1 cos α1|

|q0|

)

r t ♣♦②♥♦♠ ♥t♦♥ Pq0

s ♥ ②

Pq0(X) =

7

192X5 − 21

64X4 +

35

32X3 − 35

24X2 +

7

4

+q20

(35

768X3 − 7

16X2 +

105

64X − 35

12

)

+q40

(35

512X − 7

8

)

r q0 ≡ |raei.ns|h

qi ≡ |raei|h

♥ qi cos αi i ∈ 1, 2 r s♣② ♥ r

s ♥②t s ♥ st♠ts ♦ t rr♦r t♦ t ♣♣r♦①♠t♦♥s ♥ r ♦♠♣r t ♥②t ♥ ♣♣r♦①♠t s ♦ ∇γa

♥st t st♥ t♦ ♣♥ srt ♣♣r♦①♠t♦♥ s s ♥②

∇γas ≃ wasSsns

r t s ss♠ tt t r♥ s ♦♥st♥t ♦♥ s♠♥t s

r♥ s s t q♥t r♥ s ❬❪ ♥ t rt♦δr

h= 2 r δr

s t ♥t st♥ t♥ t♦ ♣rts ♥ h s t s♠♦♦t♥ ♥t s♦t rr♦rs r ♥ s

ǫ∇γa=

∣∣∇γ♥②ta −∇γsrt

a

∣∣∇γ♥②t

a

❲ ♦♥sr ♦♥② t ♦♠♣♦♥♥t ♦ ∇γa ♦rt♦♦♥ t♦ t ♥ r t rr♦r ♦r t srt③t♦♥ ♦ ∇γa s r② ♦♦ ♦r s ♣♦♦r srt③

t♦♥ rt♦δr

h♦r ♣♥ ss t♥ 0.1% rs t srt③t♦♥ s

s②st♠t rr♦r ♦r t ♣♣r♦①♠t♦♥ ♦ t ♦ γa ♦♥ t ♦rr ♦ 3%

❱s ♦ t ∇γa ♥t♦♥ ♥st t st♥ ♦t

❱s ♦ t rr♦r ♥t♦♥ ǫ∇γa♥st t s

t♥ ♦ t

r ♥②t ♥ r ♥ ♦♠♣t s ♥ r♥ ♦ t ♥t♦♥s ♥stt st♥ t♦ ♣♥

t srt③t♦♥ rr♦r ♦ ∇γa ♥ t ♣rs♥ ♦ ♦♠♣① ♦♥r②s s t s ♦♥ tt t rr♦r s rr ♦r♦r t rr♦r ss②st♠t s♦ tt t ♠♥t ♦ ∇γa s ②s ♥rst♠t ❲t♥ s♠t♦♥ ts s t♦ ♥♦♥♣②s ♦r ♣rts s t♦rs♦♥ ♦♥r② s♥ t rt② s ♥♦t ♥ ♦♠♣t② ② t r♣s ♦r ♣r♦♣♦rt♦♥ t♦ ∇γa

♦r♥♥ qt♦♥ ♦r γa

r♥ ♠t♦ t♦ ♦♠♣t γa ♦r ♣rt a ♥r s♦ ♦♥r② s sst t♦t t ♥ ♦r tt♦s ♣rts ♥ s tr♦r s♠♣r t♥ ♥♥②t ♦♠♣tt♦♥ ♠♥ ♦ t ♣rs♥t ♠t♦ s t♦ s ♦r♥♥qt♦♥ ♦ γa

dγa

dt= ∇γa.ua

γa = 1 ∂Ω ∩ Ωa = ∅

r t ♥t♦♥ ♦ t r♥t s ♦♠♥ t t t ttdra

dt= ua

♥♦tr ② t♦ ♦♥sr ts qt♦♥ s t♦ r♠r tt s q♥t t♦

∂γa

∂t= 0

γa = 1 ∂Ω ∩ Ωa = ∅

s ♠♥s tt t γa ♦s ♥♦t ♣♥ ♦♥ t t♠ t ♦♥② ♦♥ ♣♦st♦♥♥ s tr♦r ♥ r♥ s t s s ♠♥s t♦ ♦♠♣t γa t♦ ♦r♥t t ∇γa s sr t♦ ♦♠♣t s♥ t ♥ ①♣rss s sr ♥tr

rt♦♥ ♥ ①t♥ ♦r ♠♦♥ ♦♥rs ♥ t♦♥r② ♥ ② r ♠♦♥ ♦r♠ ♥ ts♥s tt s♠♥t ♦r tr♥ ♦♠♣♦s♥ t s ♠♦♥ t ts ♦t② t ♦♦♥ ♦r♠ s ♦t♥

dγa

dt=

∑

s∈S∇γas.u

Rs

a

γa = 1 ∂Ω ∩ Ωa = ∅

r uRs

a s t ♦t② ♦ t ♣rt a ♥ rr♥ r♠ Rs r ts♠♥t s s ① γa s ♥♦ ♦♠♣t ② s♦♥ t ♦ qt♦♥s ♦♥ t ♥♦ ♦ ∇γas ♦♠♣t r♦♠

♥t③t♦♥ ♦ t γa

♥t③t♦♥ st♣ ♦ γ0a s ♦♥ ② ♠♥ rt tr♥s♦r♠t♦♥ ♦r

♣rt ♥t② ♥①t t♦ s♦ rtr♦♥ s∣∣∇γ0

a

∣∣ > 0 ♠♦t r♦♠ ts strt♥ ♣♦st♦♥ r0

a t♦ ♥ r r t ♥t♦♥ γ (r) ≡ 1 ♦r♥st♥

ra = r0a + l

∇γ0a

|∇γ0a|

r t ♥t l s t♥ t♦ 2R R s t rs ♦ t ♦♠♣t r♥s♣♣♦rt

st ♦ t ♣r♦♣♦s ♠t♦ s s♣② ♥ r ♦r ①♠♣ tr ♣rt ♥ r s ♣ ♥ t s ♥ r t ♦

t ♦s ♥♦t ♠♦

γ s 1 ♥ s ♠♦ t♦ ts ♥t ♣♦st♦♥ ♦♥ t ♣t ♦ t r rr♦ ♣t♥ t ♦ γa t rs♣t t♦ t ♦r♥♥ qt♦♥

0.82 0.84 0.86 0.88

0

0.01

0.02

0.03

0.04

0.05

10.950.90.850.80.750.70.650.60.550.5

γa

γ (r) = 1

γ (r) < 1

r t ♦ t ♥t③t♦♥ ♦ t γ ♥①t t♦ s♦

♦t tt t qt♦♥ ♦ γa s ♥trt ♥ t♠ t s♦♥♦rr t♠♥trt♦♥ s♠ t♦ ♣r♥t s②st♠t ♥trt♦♥ rr♦rs s ➓ ♥t♦

γn+1

a = γna +

1

2

(∇

nγa + ∇n+1γa

).(rn+1

a − rna

)

t s♦ ♦♥r② s ♠♦t♦♥ss ♥r ♦r♠ ♦r ♠♦♥ ♦r♠ s

γn+1

a = γna +

δt

2

∑

s∈S

(∇

nγas + ∇n+1γas

).(uRs

a

)n+1

♦♥t♦♥ ♦♥ t t♠ st♣ s rqr t♦ ♣ t ♥trt♦♥ ♦ γa st

δt ≤ Ct,γ

1

maxa∈F ; s∈S

∣∣∣∇nγas.(uRs

a

)n∣∣∣

r Ct,γ = 0.005 r♦♠ ♥♠r ①♣r♥ s s ♥tr ♦♥t♦♥ ♥t s♥s tt t t♠ st♣ rss ♥ ♣rts ♣♦sss st ♦t② ♥♣♣r♦♥ ♦♥r② s s s♣s♥ ♥st ttr ♦♥t♦♥s ♦♥sr ♥ t♦♥ t♦ t s t♠st♣♣♥ ♦♥t♦♥s ♥ P

s ❬❪

♠r rsts

♠♥r ♥♥ ♦ tst s

♦ tst t rt♦♥ tr♠s ♥ t t sr strss ♦r♠t♦♥ ♠♥rP♦s ♦ ♥ ♦s♥♥ t ♣r♦ ♦♣♥ ♦♥rs s s♠t ♥♥ s ♠tr ♦ 1m t s♦st② ν s st t 10−1m2.s−1

s♦ tt t ②♥♦s ♥♠r s 10 s♦s tr♠ s ♠♦ t t♦rrt ♠♦ ♦ ♦rrs ♦♠♥ t t♦ ♦♠♣t t rt♦♥♦t② r s♦s tt t ♦r③♦♥t ♦t② ♣r♦ s ♥ ♦♦ r♠♥t t t ♥②t s♦t♦♥ ♥ ♥ t ♥t② ♦ t ts♠♦♥strt♥ tt t sr strss ♦rrt② ♥s t ♦② ♦r

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.20

0.2

0.4

0.6

0.8

1

10.90.80.70.60.50.40.30.20.10

ux

uxz

(m)

r P♦s ♦ ♥ ♣r♦ ♣♣ t ②♥♦s ♥♠r ♦ 10♦♦r ♦ts r♣rs♥t t ♦t② ♦ ♣rts t t st② stt rs t ♦ts • r t ♥②t ♣r♦

♦♠♣tt♦♥ ♦ t str♥ ♠♥r s ♦s s t♦ t ♦r♠ s♥ tr s ♥♦ ♥♥ ♦ t str♥ rt ♦♥ t ♦ ts s ts♦st② s ♦♥st♥t s s tst s ♦s s t♦ ♦♠♣r t ♦ t ♦♠♣t str♥ rt t♦ ts ♥②t ♥②t ♣r♦ ♦♦t② s

ux (z) = 4 Reν z

D2

(1 − z

D

)

s t♦ t ♦♦♥ ♥②t ♦ S

S (z) = 4 Reν

D2

∣∣∣2 z

D− 1∣∣∣

S(s-1)

Z(m

)

0 1 2 3 40

0.2

0.4

0.6

0.8

1

t♥r ♠♦

S(s-1)

Z(m

)

0 1 2 3 40

0.2

0.4

0.6

0.8

1

♦rrt♦♥ ♦ t♥r ♠♦

r ♦♠♣rs♦♥ ♦ t str♥ rt ♦r r♥t ♠♦s ♥ ♠♥r ♥♥♦

❲ ♥♦t ♥ r tt ♥ ts t♦rt tst s t ♦rrt ♠t♦

s stst♦r② r♣r♦t♦♥ ♦ t sr strss ♥①t t♦ t ♥t ♣r♦s ♠t♦s

t tr ♥ ♠ r ♥ t♥ t

♥ ts st♦♥ t ♣r♦♣♦s s♠ s tst ♦♥ ♠♦r ♦♠♣① ♦♠tr② ♦♥ssts ♦ s♥ ♦ ♣♣r♦①♠t② 2m ♥t ♥ 1m t t

♦ π2

rad ♥ ♥√

28

m ♦ t ♥ t ♦tt♦♠ ♠ ♦ t t♥ s♦♠tr② s ♦s♥ s♣② s♥ t trs ♦t s♦♥t♥♦s♣♦♥t ♥ s♦♣♥ ♣r♦ tsts t ♦rrt♦♥ ♦ t r♥ ♥ t ♣rs♥♦ rt② ♦♠♣rs♦♥ s ♠ t♥ r♥t ♠♦s ♥ st tr s♥ ②♥♠ s r t s♦st② ν s st t 10−2m2.s−1

t tr s

♦♠ trt♠♥ts ♦r s♦ ♦♥rs sr r♦♠ ♥ ♥t② t♦ r♣r♦ ♦rrt② st tr s r ♦♠♣r t rsts ♦t♥ ♥ t s♥s t 0.5m ♦ tr ♦r tr ss t ♥♥r♦♥s r♣s ♦rss ❬❪ t tt♦s ♣rts ♠t♦ s ❬❪ ♥ t♥ ♥ ♣r♦♣♦s ♠t♦s ①♣t t r♣s ♦rs ♣r♦ ♣♦♦r rsts s t rs ♥ t s♥s tt ♣rts ♣ s♥ ♦♥ rt t s t♦ tt tt t ♠ss♥ r ♥ t r♥ s♣♣♦rt s ♥♦t ♦♠♣♥st ♥ tst rt② s ♥♦t ♥ s♥t② ♣♦t ♦ t ♣rssr ♦ ♣rts♥st t ♣t s tr♦r ♥♦s② ♥ ② r♣r♦ ♥①t t♦ t ♦tt♦♠ tt♦s ♣rts ♠t♦ s r s ttr rsts t t♦♥t♦♥ s ♥♦t ♥sr ♥ s♦ t ♣rssr ♣r♦ s st ♥♦s② ♦r♦rts ♣♣r♦ s ♣r♦♠t t♦ sr ♥ ♦♠♣① ♦♠trs ♥ rqrst ♣rts t♦ ♠♠ t ♦♥r② ♥rs t ♦♠♣tt♦♥ ♦st ♣rs♥t ♠t♦ s s♣r♦r rsts ♥r ♣rssr ♣r♦ ♥ ♥rt ♦tt♦♠ ♥ ③r♦ ♦t② s s♦♥ ♥ r

②♥♠ s

s♠t♦♥ ♦ ♠ r t t s♠ ♦♠tr② s ♥ ♣r♦r♠ ♦r tt♦ ♦♥r② t♥qs ♣r♦s② sr ♥ t ♣rs♥t ♦♥ trs ♥t② ♦♠♥ ♦ 1m t ♥ 0.5m t ♦♥ t t♥ s ♦ ts♥ ♥ t rsts s♦♥ ♥ r ♣♣r♦s ♥sr ♠♣r♠♦♥rs t ♦t ♦ r♣s ♦rs ♠t♦ ♥ tt♦s ♣rts ♠t♦ ♥♦sr ♣rssr rtr♠♦r s♠t♦♥ t ♥r rs♦t♦♥s ♥ ♦♠♣t ② ♦♥ rs♦t♦♥ ♥ ♥ t ♥♠r ♦ ♣rts♥♣s♦ts ♦ t ♣rssr t t s♠ ♣②s t♠ r ♣♦tt ♦♥ tr

♦♠♣rs♦♥ t ❱ s♠t♦♥ ♦♥ t ♥t ❱♦♠ ♦♣♥s♦r ♦ ♣♥♦♠ ♦ t ♣rssr ♦♥ t t s ♦ t s ♣r♦♠♥ s♣② ♦♥ r

tt♦s ♣rts

♥♥r♦♥s t②♣ r♣s ♦r

Prs♥t ♠t♦

r ♦♠♣rs♦♥ ♦ t rt ♦t② ♦r st tr ♥ t♥ t ♦r r♥t ♦♥r② ♦♥t♦♥s tr 20s

♦♠♣rs♦♥ ♦ t k − ǫ ♠♦ t P ♥ ♥t

❱♦♠s s♠t♦♥ ♦ s ♣ss

♥ ♠ s t ♦♥ rr t ♦♥t♥t② ♦ t ♦ s sr♣t ♥ t♠rt♦♥ ♦ s s ♥trr♣t ♦r s♣s s s s♦♠ s♠♦♥ t ②rqrs t ss t♦ ♠rt r ♣rr ♦ rst♦r t ♠rt♦♥ ♣r♦sss ♣sss r ♥st ♦♥sst ♦ ♠♥② r♣t♥ ♠♥ts ♥ ♥ ♦♥sr s ♣r♦ ♦ ♠♥s♦♥♥ ♦ ts ♦♠♣♦♥♥ts rqrs t ♥♦ ♦ t tr♥t ♦ t♥ st♦♥ s♥ t s③ ♦t r s ts t t② ♦ t s t♦ s♠ ♣str♠

♥♥r ♥ ♦♥s r♣s ♦rs tt♦s ♣rts

Prs♥t ♠t♦ Prs♥t ♠t♦ t t s♠rs♣ srt③t♦♥

r ♦♠♣rs♦♥ ♦ t ♣rssr ♦r ♠ r tst s ♥ t♥t ♦r r♥t ♦♥r② ♦♥t♦♥s

0500100015002000

0 0.5 1 1.5 2Pressurefor e(N)

Time (s)

VOF 7 × 103 ellsVOF 3 × 104 ellsVOF 1 × 106 ellsSPH 5 × 10

3 parti lesSPH 2 × 104 parti lesSPH 8 × 104 parti les

r ♦♠♣rs♦♥ ♦ t t♠ ♦t♦♥ ♦ t ♣rssr ♦♥ t t s♦ t t♥ t ♣rs♥t P ♦r♠t♦♥ ♥ t ♥t ♦♠ ♦♣♥♦♠ t r♥t s♣ srt③t♦♥

t♦ t ♦r ♦ t ♦ t♦ s ♣ss s 3D rsr♦ ♦♥sr r 2D s♠t♦♥s r t rt rt♦♥ r ss♠t♦ ♥ ❲ r♣t r t s♠t♦♥s ♣rs♥t ♥ ❱♦ t ❬❪ ♦♠tr② ♦ t x−♣r♦ s♠t♦♥ s ♣rs♥t ♥ r rsts ♦t♥ ② P r ♦♠♣r t♦ s♠t♦♥s ♦♥ t ♦❴tr♥ ② t ♥t ❱♦♠ ♦ ♦♣ ② s ❬❪ ♠ ♦ s ♦♠♣rs♦♥ s t♦ t t ♣r♦r♠♥ ♦ P ♦r tr♥ts♠t♦♥ ♦♠♣tt♦♥ s ♦♠♣r t② s♦♥ t s♠ qt♦♥s

t ②♥♦sr r t♦s t t k − ǫ ♠♦ t t s♠

♣rssr r♥t rs♣♦♥s r♥ t ♦ ∆p

ρ∆x= 1.885m.s−2 t t t♦

r♥t ♣♣r♦s r♥♥ ♥ r♥ t t♦ r♥t srt③t♦♥♣♣r♦s P ♥ ♥t ❱♦♠

♦t② ♣r♦s t ♦t♦♥s P1 P2 ♥ P3 ♥ ♥ r r♣♦tt ♥ r rsts s♦ tt t ♠ss ♦ s ♣r♦s♦s t♦ t ♦♥s ♦t♥ t t ♣rs♥t P s♠ t s♦ ♥♦t ttt t st♥r P ♠t♦ ❬❪ t ♣rt ♦t② ♥ ② s♦st②strt♦♥s ♥♦t t t ♥t ❱♦♠ ♦♥s ❲t t ♣rs♥t ♠♦ ♦♥♥ s tt t r♠♥t s r② stst♦r②

♦♥s♦♥

♣rs♥t rt s ♣rs♥t ♥ ♣♣r♦ t♦ t s♦ ♦♥r②♦♥t♦♥ s ♦t s♠♣ ♥ r♦st s♠♣t② s ♥ t ♠♥♥r ♦♠♣t t r♥ r♥♦r♠③t♦♥ tr♠ γa t ♥trt♦♥ ♥ t♠ ♦♥② rqrs t ♦♠♣tt♦♥ ♦ ts r♥t ∇γa r♦st♥ss s t♦ t ♥trt♦♥ ♥ t♠ ♦ t ♦♥t♥t② qt♦♥ ♠s t ♥st② ♣♥ ♦♥② ♦♥ t ♣rts ♣♦st♦♥s s ♦s ♦♥ t♠ s♠t♦♥t rt② r t♠ st♣ ♥ s ♠♦r ♥t ♦r ♦♥srt♦♥♣r♦♣rts

♥t♦♥ ♦ ♥ ♦♥r② ♦rrt r♥t ♥ ♣♥ ♦♣rt♦rss s t ♦♣♣♦rt♥t② t♦ ① ♦♥r② ♦♥t♦♥s ♥ ①s ♦♥ t ♣rssr t sr strss ♥ ♥ t sr s s s k ♥ ǫ ♥ ♠♦♦ tr♥

♦r ♥♠r♦s sss st rqr ♥stt♦♥ ♥ ♦♣♠♥t ♥♠②

❱t t ♣rs♥t ♦r♠t♦♥ ♦♥ r♥t tst s s s ♣r♦tr♥t ♠♣

♣t t r♥♦r♠③t♦♥ t♦ 3D t ♠♥ ♥ s t♦ ♥ ♥♥②t ♦r♠ ♦r t ♦♠♣tt♦♥ ♦ t ♦♥trt♦♥ ♦ sr♠♥t s ♦r t ♦ ∇γa ♦ ♣rt a s ♥♦t ②∇γas ♦r ② t♦ ♦♠♣t rt② ♥ ♣♣r♦①♠t ♦ t

t② t t♦rt ♦♥srt♦♥ ♦r ♥♦♥♦♥srt♦♥ ♦ ♠♦♠♥t♠♥ ♥r ♠♦♠♥t♠ ♥ s♣② ♥ ♣r♦ ss

♦♠♥ t ♣rs♥t ♣♣r♦ t ♥♦♥♣r♦ ♥tr♥ ♦♥t♦♥s

r♥t ♦♥t♦♥s ♥ P

♦♠♣tt♦♥ ♦ t rt♦♥ ♦t② ♥ tr♥t

s

❲ ♥ ♣♣② t s♠ ♦rrt♦♥ ♦ t s♦♥ tr♠ ♦ t ♠♦♠♥t♠qt♦♥ s t ♠♥r ♦♥ ♥♦t♥ tt (µ + µT )S.n ≃ ρuτuτ ♥ t ♥t②

♥t ♦ t ♦t② tP❯

r♥t s♦st② νT t ♦❴tr♥

r♥t s♦st② νT t P❯

r♥t s♦st② νT t ♦❴tr♥

♥t ♥r② k t P❯ ♥t ♥r② k t ♦❴tr♥

r ♦♠♣rs♦♥ ♦ t k − ǫ ♠♦ t r♥♥ P ♣♣r♦♥ ♥ r♥ ♥t ❱♦♠ ♠t♦ ♥ s♠t s ♣ss

♦ ❯♥ t qt♦♥ ♥s rt♦♥ ♦t② t♦ t ♠♥♦t② ♥ t tr♥t s ♥ ♥♦tr ♥t♦♥ s♦st② s♥♦t ♦♥st♥t ♥②♠♦r ♥ s s♣♣♦s t♦ ♥r ♥ t ♥t② ♦ ♥ t ♥ s♦♥ tt t ♦t② ♣r♦ ♥ tt r s ♦rt♠s♣ ts ③♦♥ s t ♦ ②r ♦♥sr t ♣rt a t♦ ♥

♦r♠② t k = 0 ♥ t♥ νT = 0 s♦ tt r♦r t ♠♥r s ♦rt s♦s s②r r t ♠♥r s♦st② s ♠♦r ♠♣♦rt♥t t♥ t tr♥t ♦♥ ss② ♦r ♥r♦♥♠♥t ♦s r② t♥ s♦ tt ♦ ♥♦t ♥♦r k t♦ ③r♦ t

♥ ♥♥ ♦ ts ss♠♣t♦♥ s r ♥ 10% ♦ t ♥♥ ♣t

Pr♦s ♥ P1 Pr♦s ♥ P2 Pr♦s ♥ P3

r Pr♦s ♦ t ♦t② ♠♥t ♥ tr r♥t ♣♥s ♥ t s♣ss k − ǫ ♠♦ ♥ r ♥ t k − ǫ ♠♦ t ♦❴tr♥ ♥r♥

t ♦ ②r ♦ s♠♦♦t uτ ♥ ♦t♥ r♦♠ t ♦♦♥ t ♥ trt ♦rt♠

|ua|uτa

=1

κln(zauτa

ν

)+ 5.2

♦ qt♦♥ ♠st r② tt t ♥♦♥♠♥s♦♥ st♥ t♦ t zauτa

νs rtr t♥ 11 ❲ ♦ s♦ s ♦ s ♦r r♦ s ♦r s

s ♦ ♦t ♥ t ♠♥r ♥ t ♦ ②r s s rs t t♥ t k − ǫ ♠♦ ♠st s♦ ♠♦ ♦r ♦ ②♥♦s ts♦r ♠♦r ♥♦r♠t♦♥ s ❬❪

❱♦t② t t

❲ ♦sr ♥ t ♦r♠ tt t ♦t② t t us s ♦♥sr♥ t ♦♥r② tr♠ ♦r♠② t ♥♦s♣ ♦♥t♦♥ ♦ ♠♣♦s tt t♦t② t t s t ♦t② ♦ t ts 0 ♦r ♠♦t♦♥ss s s ♠♣♦s ♦r ♠♥r ♦ ♥ t tr♥t s t s ♣rr t♦ ♥♦t♦ s♦ t s♦♣ ♦ t ♦t② ♣r♦ s ♠ rr t t t♥ ♥ t♦ ②r r ♣rts r ss♠ t♦ s ♥t t♦ trt② Sa ♥①t t♦ t ♥ t♦ ♥tr♣♦t t ♦t② t t ♦ ♦ s♦ t t ♦t② ue ♦ ♣rts ♥ t s♦s ♥ rt♦♥tr♠s

due

dt=

1

γe

∑

b∈Fmb

µTe + µTb

ρeρb

ueb

r2eb

reb.∇web

︸ ︷︷ ︸s♦s tr♠

− 2uτeuτe

γe

∑

s∈S|∇γes|

︸ ︷︷ ︸rt♦♥ tr♠

❲ ♥♦t r tt qt♦♥ s t ♠♦♠♥t♠ qt♦♥ ♣♣ t♦ ♥ ♣rt t ♥tr rt② ♥♦r ♣rssr r♥t ♥ ♥ us t♦ t r t♥ ♣rts e1 ♥ e2 ♥ ♥ ➓ stt t♦ ♦ ♥ ♦ s♣ ♦t② t t ♥ ②♥♦s ♥♠rs♠t♦♥ s s♦ s ♥ ♠♥② ♦s s s ♥ ♥t♠♥ts s r♦t ❬❪ ♥t② t♦ r ♥ ♠♥ tt t ♣rts ♥ E r ♥ t r♥ ♣♦♥ts ♥ ♦ ♥♦t ♠♦ t t ♦t② ue

t t t ♦t② t s t♦ s② ♦♥② s t ♦t② ue t♦ ♣ts♦s ♦rs ♦ ♣rts ♥trt♥ t t ♥ t♦ ♦♠♣t tstr♥ rt S

① ♦♥t♦♥s ♦♥ t ♥t ♥r②

♣♥ ♦♣rt♦r ♣♣ t♦ t tr♥t ♥t ♥r② rs t♦

1

ρa

Lapa

(µb +

µTb

σk

, kb)

=1

γa

∑

b∈Fmb

2µ + µTa/σk + µTb/σk

ρaρb

kab

r2ab

rab.∇wab

s t s ss♠ tt tr s ♥♦ ① ♦ k r♦♠ t ♦♥r② ∂k

∂n=

0 t t qt♦♥ s♦ s ♥ ♣ ♦ Lapa ♥ ♣②s ♠♥♥ s tt t tr♥t ♥t ♥r② s ♦♥② rt ② t♠♥ ♦ ♦r♦r ♥st ♦ s♣②♥ ♦♥r② ♦♥t♦♥s t t t ♥t♦♥ ♣♣r♦ srs t ♥ ♦ tr♠s ♥ t ♥t② ♦

t r t s ss♠ tt P = ǫ ts ♠♣s tt t ♦♥t♦♥∂k

∂n= 0

s ♥♦t ♦♥② t t t ♥ t ♦ ♥t② ♦ t s♦ ♦♥r② ss♠ t ♦ t♦ ② tr♥t tt s t♦ s② t t♥

s♦s s②r ♥ s ②♥♦s ♥♠r s ♦r k − ǫ ♥ t ♦ ♥♦t s♦ t k− ǫ ♠♦ ♣ t♦ t r k s t♦rt② ①♣t t♦ 0 t ♣ t♦ s♠ st♥ δ r♦♠ t r t tr♥ s ②sts νT ≫ ν ♠♥ ♥t ♦ t ♣rs♥t r♥♥ ♣♣r♦♦♠♣r t♦ ♥ r♥ ♦♥ s tt t r♥♥ ♣rts ♥ F \ E t st t st♥ ♦ t ♦rr ♦ δr r♦♠ ♥② t s ♦♥ ♦t ♠♥ ♥ts ♦♠♣r t♦ t ♠ss ♠t♦s r t ♦♥r②♠♥ts r s♣♣♦s t♦ t rt st♥ t♦ t t r ♦♥②♦s ♦♥ ♣rts r t② t ♥♦♥③r♦ st♥ r♦♠ t s

♦ st♠t k t t ♥ t s ♦r t ♥st② t♣rssr ♦r t sr strss

ke =1

αe

∑

b∈F\EVbkbwbe

❲ ♥♦t tt ts ♣♣r♦①♠t♦♥ s ♦♥sst♥t t t ss♠♣t♦♥∂k

∂n= 0

♥②

ks =ke1 + ke2

2

① ♦♥t♦♥s ♦♥ t ss♣t♦♥ ♦ ♥t ♥r②

♣♥ ♦♣rt♦r ♣♣ t♦ ǫ rqrs t ♦r ∂ǫ/∂n ♥♥ ss♠ tt t ♦ s ② tr♥t t♥ r② s♥ ♣rt♥ t r ♦ ♥♥ ♦ ∃s ∈ S/ |∇γas| > 0 s ♥ t ♦ ②rr

k ≃ u⋆2

√Cµ

ǫ =u⋆3

κz

νT = κu⋆z

t qt♦♥s ♥ r r♦♠ t qr♠ P = ǫ

r z s t st♥ t♦ t ♣rt a s ♥trt♥ t sr s stt z = max (ras.ns; δr) r δr s t ♣rt r ♥t s♣♥κ s t ❱♦♥ r♠♥ ♦♥st♥t t t ♦ 0.41 ♥ u⋆ s rt♦♥ ♦t②♠sr♥ t tr♥

u⋆s =

√ks

C1

4

µ

♥ r♦♠ qt♦♥ ♥ ♦r t ① ♦ ǫ

νTa

σǫ

∂ǫa

∂ns

= − 2u⋆4s

σǫκδras

t♦r 2 s ♣r♦ ② rst♦rr ♣♣r♦①♠t♦♥ t ① s tt t st♥ z

2 s r② s ♥ s♣② r s ǫ s s♣♣♦s

t♦ r② s 1zr z s t st♥ t♦ t

s t ♣♥ ♦♠s

1

ρa

Lapa

(µb +

µTb

σǫ

, ǫb)

=1

γa

∑

b∈Fmb

2µ + µTa/σǫ + µTb/σǫ

ρaρb

ǫab

r2ab

rab.∇wab

+4

γaρa

∑s∈S |∇γas| ρs

u⋆4s

σǫκδras

s s ♥ ♣ ♦ Lapa ♥

r♥s

❬❪ sr♠ ♦♥t s ❲ Pr♦t rt♦♥ ♦r♠t♦♥s ♦♥tt ♦rt♠ ♦r r ♦♥rs ♥ t♦♠♥s♦♥ s♣ ♣♣t♦♥s

❬❪ r ♦r♥ ss♥r♥ rr♥t P ♥ ♠♣r♦ s♣ ♠t♦♦rs r ♦rr ♦♥r♥ ♦r♥ ♦ ♦♠♣tt♦♥ P②ss

❬❪ ♦♥♦ ♥♥t t t ♥♥ ♦ s♠♥②t ♣♣r♦ ♦r s♣ ♠♦♥ ♦ s♦ ♦♥rs4th P ♦rs♦♣ ♥ts r♥

❬❪ ♦♥♥ tr ♣ ♣rt ♦♥r② ♦rs ♦r rtrr② ♦♥rs ♦♠♣tr P②ss ♦♠♠♥t♦♥s

❬❪ r♦♥ ♦ r♦ ♦ ♠ ♥♠r ♥♠r s♠s ♦rt s♣ ♠t♦ ♣♣t♦♥ ♥ r sr ♦s ♥ ♣t♦♥ tr♥s 4th

P ♦rs♦♣ ♥ts r♥

❬❪ ♦③é ss♥r♥ ♦r♠ ① ♠t♦ t t ♦♥r② ♦r s♣ 4th P ♦rs♦♣ ♥ts r♥

❬❪ ♦♥♥ ♠t♥ r sr ♦s t P ♦r♥ ♦ ♦♠♣

tt♦♥ P②ss

❬❪ ❱♦ ss ♠r ♠♦♥ ♦ ♦♠♣① tr♥t rsr♦s t t s♣ ♠t♦ ♥ ♦r ♥tr♥t♦♥ ♦r♥ ♦r

♠r t♦s ♥ s ♦ ❯tt♣①♦♦r

❬❪ ❱ P ❲t ♣rt ♣r♦s tr♠♠t♦s♥①t tr♠ ♥ s♠♦♦t♣rt ②r♦②♥♠s t ♦s t ♣♣

❬❪ ♠♥ ♦♥t ②♥♠ r♥♠♥t ♥ ♦♥r② ♦♥tt ♦rs ♥s♣ t ♣♣t♦♥s ♥ ♦ ♣r♦♠s ♥t ♦r♥ ♦r ♠r

t♦s ♥ ♥♥r♥ sr t♦s ♥t♥s ♥ ♣♣t♦♥s

❬❪ ♦♥♥ ♠♦♦t ♣rt ②r♦②♥♠s ♥♥ r ♦ str♦♥

♦♠② ♥ str♦♣②ss

❬❪ ♦rrs P ♦① P ❩ ❨ ♦♥ ♦ r②♥♦s ♥♠r ♥♦♠♣rss♦s s♥ s♣ ♦r♥ ♦ ♦♠♣tt♦♥ P②ss

❬❪ ❱♦ ♥s ♦ ② ♦♠♣rss ♦s ♥ ts ♣♣t♦♥

t♦ t P ♠t♦ t♦ ♣s

❬❪ ♥r ♣♥ t♠t ♠♦s ♦ tr♥ ♦♥♦♥♠ Prss

❬❪ rr♥ r♥ ♦rs ❱♦ ♠♣r♦ t♠ s♠ ♥trt♦♥ ♣♣r♦ ♦r ♥ t s♠ ♥②t ♦♥r② ♦♥t♦♥s♥ s♣rts Pr♦ V th P ♥tr♥t♦♥ ❲♦rs♦♣ ♥str❯

❬❪ ♦st♥ P♦♦ ♦ ♦♥srt♦♥ ♦ t ♠t♦♥♥ s♦♥❲s② ♥ r♥s♦ 3rdedition

❬❪ r♦♥ ♦ Pr♥s♦♥ ♠r s♠t♦♥ ♦ t ♦ ♥ Pt♦♥ tr♥ s♥ t ♠sss ♠t♦ P ♥ ♥ s♠♣ s♦♦♥r② trt♠♥t t r♦♣♥ ♦♥r♥ ♦♥ r♦♠♥r②

②♥♠s ♥ r♠♦②♥♠s t♥s r

❬❪ ❲♥♥ Ps ♣♦②♥♦♠ ♣♦st ♥t ♥ ♦♠♣t② s♣♣♦rt r ♥t♦♥s ♦ ♠♥♠ r ♥s ♥ ♦♠♣tt♦♥

t♠ts ♠r

❬❪ ❱♦ ss ♥♠♦ ♦r ♦rt ♦♥ s ♣ss t s♣ ♥ r♥ ♦s t ♥♥♦ tr♥t ♦sr Pr♦ IIIrd P ♥tr♥t♦♥ ❲♦rs♦♣ s♥♥ ss

❬❪ ♦❴tr♥ ♥t ❱♦♠ ♦ ♦r t ♦♠♣tt♦♥ ♦ r♥t

♥♦♠♣rss ♦s ♥str ♣♣t♦♥s ♦ ♥tr♥t♦♥ ♦r♥♦♥ ♥t ❱♦♠s

❬❪ P♦♣ r♥t ♦s ♠r ❯♥rst② Prss

❬❪ r♦t ②r♦②♥♠s ♦ r r ♦s ♦♥ t t

♥t ♠♥t t♦ ♦♥ ❲②